1. Field of the Invention
This invention relates generally to the simulating and analysis of surface acoustic wave devices and, more particularly, to a method and a system of simulating a surface acoustic wave on a simulated structure.
2. Description of the Related Art
A Surface Acoustic Wave (SAW) is a standing or traveling acoustic wave on the surface of a substrate. A typical SAW device includes a substrate (typically made from a piezoelectric material) and a periodic array of electrodes on the surface of the substrate. Piezoelectric materials deform in response to a voltage being applied to them. Piezoelectric materials also generate a voltage in response to stress being applied to them.
A SAW device may be simulated by numerically solving governing equations which describe the behavior of the device. Examples of such governing equations are Newton's equation of motion and Gauss' equation of charge conservation. The material properties, geometry and driving voltages are very important to simulating the behavior of the SAW device.
One method of simulating a SAW device is to use the Finite Element (FE) method to solve the governing equations. The FE method involves creating a mesh, in which a problem domain is divided into a set of discrete sub-domains called elements. The governing equations, which describe the behavior of each element, are then solved for each element. The governing equations are typically solved numerically. The size of the mesh will determine the amount of computational time required to simulate the SAW device. The mesh elements should be small enough to effectively simulate the behavior of the SAW device, but not so small as to require an unreasonable amount of computational resources.
The typical SAW device includes a thin electrode and a relatively thick substrate. As the frequency of the waves being simulated becomes higher, the coupling of the SAW and the bulk acoustic wave (BAW) becomes so intense that distinguishing the SAW from the BAW becomes an issue.
An additional method of simulating a SAW device is to use a hybrid FE (HFE) method. A typical HFE method when used to simulate a SAW device would use the FE method in a region of the electrodes including a portion of the substrate and an analytic method for the remaining region of the SAW devices substrate. Examples of analytic methods that have been used include: a Periodic Green's Function, a Boundary Element Method (BEM) or a Spectral Domain Method (SDM). In the past these approaches have assumed a semi-infinite substrate. Thus, the thickness of the SAW device is not fully taken into account using these methods.
An example of such a HFE method was described by Koji HASEGAWA et al., in Hybrid Finite Element Analysis of Leaky Surface Acoustic Waves in Periodic Waveguides, Japanese Journal of Applied Physics, Part 1, 35(5B): 2997-3001, 1996, The Japan Society of Applied Physics, Tokyo, JAPAN (hereinafter Hasegawa).
Hasegawa describes a method in which an inhomogeneous region including the electrodes is analyzed by the FE method and the substrate region, which is assumed to be semi-infinite, is approximated by an expansion of space harmonics. Only partial roots of Christoffel equations for each space harmonics are used for the basis of the expansion.
However, if Hasegawa is modified such that the substrate is assumed to be finite and all the roots of the Christoffel equations are used, floating-point overflow issues arise. So the numerical solution is limited to cases in which the substrate thickness is less than a few (i.e., three) wavelengths or infinite.
These limitations in Hasegawa and methods of this type leave a large swath of problems for which finding a numerical solution is difficult. For example these methods do not take into account the thickness of the substrate or the effect of reflections from the bottom of the substrate.
In view of the shortcoming described above there is a need for systems and methods to address the shortcomings of the above methods while maintaining high accuracy and low computational costs.